Novel source term and Lorentz Torque

Larmour's formula (which is the reason we say electron should radiate) is derived from Poynting's theorem, whose starting point is

$$P=\textbf{F}\cdot\textbf{v},$$
But we know electron has magnetic moment!
Therefore a new term is added
$$P=\textbf{F}\cdot\textbf{v}+\boldsymbol{\tau}\cdot\boldsymbol{\omega}$$
which is ignored by Poynting's theorem.
​Since
$$\boldsymbol{\tau}=\textbf{m}\times\textbf{B},$$
we have
$$P= q \mathbf{v} \cdot \mathbf{E} - (\textbf{m}\times\boldsymbol{\omega})\cdot \textbf{B},$$
which is
$$P=\int_V (\mathbf{J} \cdot \mathbf{E}-\mathbf{K} \cdot \mathbf{B}) ~\mathrm d^{3}x,$$
Just like the first term says `electricity in motion is a source J for a new field (magnetic field)', the new term tells us `magnetism in rotation is a source K for a new field', where
$$\textbf{K}=\textbf{M}\times\boldsymbol{\omega},$$

is the new source term. 

$$\boldsymbol{\tau}=(\textbf{m}+\textbf{v}\times\textbf{p})\times\textbf{B}$$